Do you insure your horses? Which insurance do you recommend to choose?

I am in France, what works here. Maybe it will help.
For starters, you should know that there are two types of insurance:
- Third party liability insurance (IRMI, Investopedia),
- Mortgage-theft-veterinary insurance (Petplan Equine,Gow Gates).
In France, the third party liability insurance covers the damage caused by the horse to a third party. There are also two cases to be distinguished: "in action of horsemanship", it is the license FFE that works, whether you own or not your mount; And "hors d'action équitation", it is the RCPE (Responsabilité Civile Propriété d'Equidé, the equivalent outside France is Civil Liability Ownership of Equine) that works.
In France, the mortgage-theft-veterinary insurance which concerns the "damage" that could undergo the equine. The bottom line is mortality-theft, and then we add the guarantee veto, disability, and other options proposed by the insurers.
It is not compulsory to take out insurance when buying a horse. But in my opinion, it's better to have one, for the simple reason that horses are living beings just like us and that it can all happen at any time!
There is ample insurance for horses: KBIS, Markel insurance, Petplan Equine,…. It's up to you to see which is best for you, budget but also according to what you do with your horse.